微分几何是运用微积分的理论研究空间的几何性质的数学分支科学。微分几何与拓扑学等其他数学分支有紧密的联系，对物理学的发展也有重要影响。A.麦肯纳利著的《微分几何初步（英文版）》是一部为掌握数学基础知识之后继续领略高等数学之美的本科生而编写的标准教科书，各章有习题。

1 Basic Objects and Notation
1.1 Sets
1.2 Functions
2 Linear Algebra Essentials
2.1 Vector Spaces
2.2 Subspaces
2.3 Constructing Subspaces I: Spanning Sets
2.4 Linear Independence, Basis, and Dimension
2.5 Linear Transformations
2.6 Constructing Linear Transformations
2.7 Constructing Subspaces II: Subspaces and Linear Transformations
2.8 The Dual of a Vector Space, Forms, and Pullbacks
2.9 Geometric Structures I: Inner Products
2.10 Geometric Structures II" Linear Symplectic Forms
2.11 For Further Reading
2.12 Exercises
3 Advanced Calculus
3.1 The Derivative and Linear Approximation
3.2 The Tangent Space I" A Geometric Definition
3.3 Geometric Sets and Subspaces of Tp(Rn)
3.4 The Tangent Space II: An Analytic Definition
3.5 The Derivative as a Linear Map Between Tangent Spaces
3.6 Diffeomorphisms
3.7 Vector Fields: From Local to Global
3.8 Integral Curves
3.9 Diffeomorphisms Generated by Vector Fields
3.10 ForFurther Reading
3.11 Exercises
4 Differential Forms and Tensors
4.1 The Algebra of Alternating Linear Forms
4.2 Operations on Linear Forms
4.3 Differential Forms
4.4 Operations on Differential Forms
4.5 Integrating Differential Forms
4.6 Tensors
4.7 The Lie Derivative
4.8 For Further Reading
4.9 Exercises
5 Rlemannlan Geometry
5.1 Basic Concepts
5.2 Constructing Metrics; Metrics on Geometric Sets
5.3 The Riemannian Connection
5.4 Parallelism and Geodesics
5.5 Curvature
5.6 Isometrics
5.7 For Further Reading
5.8 Exercises
6 Contact Geometry
6.1 Motivation I: Huygens' Principle and Contact Elements ...
6.2 Motivation II: Differential Equations and Contact Elements
6.3 Basic Concepts
6.4 Contact Diffeomorphisms
6.5 Contact Vector Fields
6.6 Darboux's Theorem
6.7 Higher Dimensions
6.8 For Further Reading
6.9 Exercises
7 Symplectic Geometry
7.1 Motivation: Hamiltonian Mechanics and Phase Space
7.2 Basic Concepts
7.3 Symplectic Diffeomorphisms
7.4 Symplectic and Hamiitonian Vector Fields
7.5 Geometric Sets in Symplectic Spaces
7.6 Hypersurfaces of Contact Type
7.7 Symplectic Invariants
7.8 For Further Reading
7.9 Exercises
References
Index